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Journal of Physics Condensed Matter

Return to coherence via Debye–Waller factor quenching

A.C. Levi

Published Sep 14, 2009• 10 citations
Publisher

Abstract

A number of scattering situations can naively be expected to show essentially classical features, because of a (generalized) Debye-Waller factor causing decoherence and destroying interference and diffraction phenomena. It is shown, however, that such decoherence may be quenched in several ways: A-the momentum transfer to a large system may be separated into subtransfers to individual atoms, each subtransfer being too small to cause a strong Debye-Waller effect; B-(more significantly) the high-frequency vibrations may be ineffective in causing strong decoherence, because the corresponding correlation functions remain non-vanishing for too short a time. When this Debye-Waller factor quenching takes place, coherence is restored and typical quantum wave-like phenomena reappear.

DOI
10.1088/0953-8984/21/40/405004

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A number of scattering situations can naively be expected to show essentially classical features, because of a (generalized) Debye-Waller factor causing decoherence and destroying interference and diffraction phenomena. It is shown, however, that such decoherence may be quenched in several ways: A-the momentum transfer to a large system may be separated into subtransfers to individual atoms, each subtransfer being too small to cause a strong Debye-Waller effect; B-(more significantly) the high-frequency vibrations may be ineffective in causing strong decoherence, because the corresponding correlation functions remain non-vanishing for too short a time. When this Debye-Waller factor quenching takes place, coherence is restored and typical quantum wave-like phenomena reappear.
This note has no new results and is therefore not intended to be submitted to a "research" journal in the foreseeable future, but to be available to the numerous individuals who are interested in this issue. The Debye-Waller factor is the ratio of the coherent scattering or absorption cross section of a photon or electron by particles bound in a complex system to the value for the same process on an analgous free particle. It is often interpreted also as the probability of the coherent process, normalized to unity, with the difference between unity and the Debye Waller factor interpreted as the probability of incoherent processes. The Debye-Waller factor is then interpreted as a measure of decoherence. The breakdown of this description for a test particle which cannot give or lose energy is not generally appreciated. Prime examples are: Bragg scattering, the Mössbauer effect and related phenomena at zero temperature. The physics of the change in the interpretation of the Debye-Waller factor is summarized here in a hopefully pedagogical manner.
The famous Nils Bohr's quantum-classical correspondence principle states that the classical mechanics is a limiting case of the more general quantum mechanics. This implies that ``under certain conditions" quantum laws of motion become equivalent to classical laws. One of the conditions is fairly obvious: the corresponding classical action should be very large as compared with the Planck's constant. But this condition is not sufficient. Suppression of effects of quantum interference ("decoherence") is the phenomenon of primary importance for understanding of the Bohr's principle. Being, in essence, of quite general nature, this problem takes on special significance in the case of non-linear classically chaotic quantum systems. Whereas the rapid decay of phase correlations is an underlying feature of the classical dynamical chaos, the "quantum chaos" by itself is not capable of destroying the quantum phase coherence. Strictly speaking, any initially pure quantum state remains pure during an arbitrary long unitary evolution. Basically, formation of incoherent mixed states (decoherence) results only from the process of preparation of a mixed initial state or interaction with a noisy environment. Just the decoherence brigdes the quantum and classical worlds. Peculiarities of the time behavior of Peres fidelity, state purity, Shanon and von Neumann entropies are analyzed in detail. We demonstrate the ways the decoherence shows up in periodically driven systems that can be associated with Ramsey-type interferomentry experiments with ion traps. Finally, decoherence in ballistic electron quantum transport caused by interaction with a disordered environment is considered.
The Debye-Waller factor may be close to one, hence ineffective for decoherence, if it is caused by high-frequency vibrations. This appears as a manifestation of the Zeno effect, where the quantum properties of a system do not deteriorate (or do so very slowly) if the system is subjected to frequently repeated interactions. In the case of atom-surface scattering, the decoherence time is shown to be long for high frequencies, such that the oscillation period is short in comparison to the duration of a collision. For a periodic perturbation the restoration of coherence, in the case where many oscillation periods take place during a prescribed time, is shown.
Decoherence is a relatively recent concept in the field of quantum mechanics. Although the pioneers of the field must have understood that loss of phase coherence in quantum superpositions is the reason underlying the appearance of definite outcomes in the quantum measurement problem, the latter was not treated in terms of decoherence until sixty years after the formulation of quantum mechanics as described in a report by Joos and Zeh in 1983 [1]. However, soon after, the theory was developed in further detail, and experiments to measure the actual decoherence rates in various systems commenced. Today, decoherence is the main concern for another reason, i.e., the fact that superposition states must be maintained undisturbed in the quantum communication systems and decoherence presents a strong limitation for their practical application. Decoherence appears in open quantum systems, where the basic system under consideration interacts relatively strongly with an environment. Decoherence times may be as long as seconds for small atomic systems in extreme vacua, although below what is presently measurable (i.e., less than a fraction of a femtosecond) in most liquids and solids, where coupling to the surrounding molecules or atomic arrangements is strong. The relatively slow decoherence in well-isolated particle systems has been described in several recent articles on quantum optics, while the rapid decoherence in liquids and solids has not received this kind of attention. It is the purpose of the present review to fill this gap.
The study and control of coherence in quantum systems is one of the most exciting recent developments in physics. Quantum coherence plays a crucial role in emerging quantum technologies as well as fundamental experiments. A major obstacle to the utilization of quantum effects is decoherence, primarily in the form of dephasing that destroys quantum coherence, and leads to effective classical behavior. We show that there are universal relationships governing dephasing that constrain the relative rates at which quantum correlations can disappear. These lead effectively to speed limits which become especially important in multipartite systems.
We develop a new method for the calculation of the atom scattering analog of the Debye–Waller factor. Unlike x-ray and neutron scattering the properties of an atom scattered by a solid surface cannot be computed by perturbation theory; therefore the simple Debye–Waller theory cannot be applied. Nevertheless the Debye–Waller phenomenon has a close analog: the elastic intensity is depressed due to the uncorrelated part of the thermal motion of the lattice atoms. To compute this effect we develop a time dependent scattering theory in which the quantum properties of the scattered atom are described by propagating coherently an ensemble of wave packets and lattice motion is simulated by a classical Langevin equation. Applications are made to He and Ne scattering from a surface whose lattice dynamics mimics that of Pt(111) but whose corrugation was slightly increased to enrich the diffraction structure.